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Chapter 4: Problem 30

Find the maximum or minimum value for each function (whichever is appropriate). State whether the value is a maximum or minimum. $$y=-\frac{1}{3} x^{2}-2 x$$

Short Answer

Expert verified
The maximum value is -9 at x = 3.

Step by step solution

01

Identify the Type of Function

The given function is a quadratic function, written in the form of a downward-opening parabola: \[ y = -\frac{1}{3}x^2 - 2x \]Since the coefficient of \(x^2\) is negative, the parabola opens downwards, indicating that the function has a maximum value.
02

Find the Vertex of the Parabola

The vertex of a quadratic function \(ax^2 + bx + c\) gives the maximum or minimum point. The x-coordinate of the vertex can be found using the formula:\[ x = -\frac{b}{2a} \]Here, \(a = -\frac{1}{3}\) and \(b = -2\). Substituting these values, we get:\[ x = -\frac{-2}{2(-\frac{1}{3})} = \frac{2}{\frac{2}{3}} = 3 \]
03

Calculate the Corresponding y-Value

Substitute \(x = 3\) into the original function to find the corresponding y-value (the maximum value of the function):\[ y = -\frac{1}{3}(3)^2 - 2(3) \]\[ y = -\frac{1}{3}(9) - 6 \]\[ y = -3 - 6 = -9 \]
04

Conclusion

Therefore, the maximum value of the function \(y = -\frac{1}{3}x^2 - 2x\) is \(-9\) at \(x = 3\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parabola
A parabola is a U-shaped curve that can open either upwards or downwards. It is the graphical representation of a quadratic function. Quadratic functions have the general form:
  • \( y = ax^2 + bx + c \)
In this expression, the coefficient \( a \) determines the direction the parabola opens. If \( a \) is greater than zero, the parabola opens upwards. If \( a \) is less than zero, as in our example, the parabola opens downwards.
The symmetry in parabolas means that they have a single line of symmetry, which passes through the vertex, making it an essential point for finding the minimum or maximum value.
Vertex of Quadratic Function
The vertex of a quadratic function is a critical point that tells us either the maximum or minimum value of the function. The vertex can be thought of as the "tip" of the parabola. For a function written as \( y = ax^2 + bx + c \), the x-coordinate of the vertex can be found by using the formula:
  • \( x = -\frac{b}{2a} \)
This formula helps determine where the vertex lies along the x-axis. In our function \(-\frac{1}{3}x^2 - 2x\), plugging in \( a = -\frac{1}{3} \) and \( b = -2 \) gives us an x-value of 3 for the vertex.
After finding \( x \), substitute it back into the function to discover the y-coordinate. This point \((x, y)\) is the vertex of the parabola, indicating its highest or lowest point.
Maximum Value
The maximum value of a function is the highest point the function reaches.
Since the function \( y = -\frac{1}{3}x^2 - 2x \) opens downwards, it has a maximum value rather than a minimum. The vertex of the parabola is the point where this maximum occurs.
For this specific function, you can find the maximum value by first determining the vertex's x-coordinate, \( x = 3 \), and then substituting it back into the function to find \( y \), which in this case results in \( y = -9 \).
Thus, the maximum value of the function is \( -9 \), occurring at the vertex when \( x = 3 \).
Downward-Opening Parabola
A downward-opening parabola is one that curves downward, forming an upside-down U-shape. This configuration occurs when the coefficient \( a \) in the quadratic function \( y = ax^2 + bx + c \) is negative.
In the quadratic function \( y = -\frac{1}{3}x^2 - 2x \), \( a = -\frac{1}{3} \), which is less than zero, confirming that we deal with a downward-opening parabola.
This means that any value of \( x \) will result in \( y \) being lower than the function's maximum value. These properties are important for understanding the behavior of quadratic functions and graphing them appropriately.

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Most popular questions from this chapter

Sketch the graph of each rational function. Specify the intercepts and the asymptotes. $$y=-1 /(x+2)^{3}$$

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